I've been saving this post for March 14th, but I decided that I couldn't wait that long! At times, I am absolutely fascinated by mathematics. International Pi Day occurs every year on March 14th. Mathematics enthusiasts the world over celebrate in their own special way. For example, several years ago, my daughter and I toured Rose-Hulman Institute of Technology on a college visit. Rose-Hulman has one of the best engineering programs in the country. The visit happened to occur on International Pi Day, and the tour guide was absolutely thrilled to share with us that "someone" the night before wrote down in chalk all the digits of Pi on all of the sidewalks around campus. My wife is a middle school math teacher and always holds a competition to see which student has memorized Pi to the most decimal places - oh, and they always eat pie too!
Just in case you didn't know, Pi (denoted by the symbol π) is a constant that is the mathematical ratio of a circle's circumference to its diameter. It is an irrational number, which means that it cannot be expressed exactly as a ratio of two integers (the ratio 22/7 comes awfully close, but not close enough). As a result, its decimal representation keeps going on and on and on without ever repeating. The decimal digits appear to be randomly distributed, though this has never been proved to the satisfaction of most mathematicians.
Well, I was "today years old" when I first learned that there is a sequence of six consecutive nines starting at the 762nd decimal place of Pi. It's actually a HUGE coincidence - some would say it's a statistical anomaly (more on that below). The world renowned physicist Richard Feynman allegedly once said that he used to memorize all of the decimal points of Pi up to this sequence, after which he would say "and so on!" Subsequently, the 762nd decimal place became known as "Feynman's Point".
The author Douglas Hofstadter wrote in his book Metamagical Themas:
"I myself once learned 380 digits of Pi, when I was a crazy high-school kid. My never-attained ambition was to reach the spot, 762 digits out in the decimal expansion, where it goes "999999", so that I could recite it out loud, come to those six 9's, and then impishly say, "And so on!"
What's simply amazing is that the probability of having a string of six 9's in a completely random sequence of numbers is exceedingly low - some mathematicians have calculated the probability to be 0.08%. I have no idea how they came up with that probability, so for now we will just have to trust my sources. I suppose it has something to do with what is known as the infinite monkey theorem, which states that if you let a monkey hit the keys on a typewriter an infinite number of times, at some point the monkey will type out the entire works of William Shakespeare (for a humorous anecdote, check out the article "The Infinite-Monkey Theorem: Field Notes" in The New Yorker). Well, infinity is a long time, so I guess that a monkey could do that, right?
Apparently there is a relatively straightforward proof to the infinite monkey theorem. I won't bore you with the details, but it only requires a basic understanding of the laws of probability and some basic mathematics. The point is that the probability of a monkey randomly typing out even a short passage from any of the plays of Shakespeare is exceedingly small, but it's not zero! In other words, given enough time, anything is possible!
What's absolutely profound, in my opinion, is that something that is as ordered as a sequence of six 9's can appear out of the blue in the infinitely long number of decimal places of Pi. As Adam Frank wrote several years ago in an online post for NPR ("The Infinite Monkey Theorem Comes to Life"), "The emergence of such intricate complexity from randomness is counterintuitive to brains that have evolved to see pattern and meaning everywhere. To digest the true significance of the infinite monkey theorem, it's best to turn from science to art..."
I've been spending my "Summer Break" reading a lot about chaos theory, complexity science, and complex adaptive systems, all topics that I want to explore further in upcoming posts. For now, back to Adam Frank, "Order emerging from chaos, meaning emerging from randomness, right before your eyes and not a banana in sight." And it all started with Pi.
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